Finite Group Databasebeta

← #134063

Subgroups — #134063

Order 512.

804 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 804 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2499C211yesmaximal, characteristic
2#11512C212yesmaximal, complemented
3#11511C212yesmaximal, complemented
4#8555C212yesmaximal
5C22×C22⋊Q8C2211yescharacteristic
6C22×#117C411yescharacteristic
7#2499—24no—
8#2499—24no—
9C2×#397—22no—
10C2×#391C411yescharacteristic
11C2×#391C411yescharacteristic
12C2×#387—22no—
13C2×#384C2211yescharacteristic
14C2×#382C2211yescharacteristic
15C2×C4×C22⋊C4C411yescharacteristic
16#1821—22no—
17#1463C2214yes—
18#1462—22no—
19#1325—24no—
20#1324—24no—
21C23×Q8C2×C411yescharacteristic
22C24×C4C2×C411yescharacteristic
23C2×C22⋊Q8—22no—
24C2×C22⋊Q8—22no—
25C2×C22⋊Q8—42no—
26C2×C22⋊Q8—42no—
27C2×#117—22no—
28C2×#117—22no—
29C2×#117—42no—
30C2×#117—42no—
31C22×C4⋊C4—22no—
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C4⋊C4D412yes—
35C22×C4⋊C4C2311yescharacteristic
36C22×C22⋊C4—21no—
37C22×C22⋊C4—21no—
38C22×C22⋊C4D412yes—
39C22×C42—21no—
40C22×C42C2×C411yescharacteristic
41C22×C42C2×C411yescharacteristic
42#407—24no—
43#397—216no—
44#391—22no—
45#391—22no—
46#391—22no—
47#391—22no—
48#387—216no—
49#384—22no—
50#384D414yes—
51#382—22no—
52#382D414yes—
53C4×C22⋊C4—22no—
54C4×C22⋊C4—22no—
55C2×#95—22no—
56C2×#95—22no—
57C2×#95—21no—
58C2×#95—22no—
59#343—44no—
60#324—24no—
61#324—24no—
62#323—44no—
63C25C22⋊C411yescharacteristic
64C22×Q8—21no—
65C22×Q8—21no—
66C22×Q8—41no—
67C22×Q8—41no—
68C22×Q8—42no—
69C22×Q8—42no—
70C23×C4—22no—
71C23×C4—21no—
72C23×C4—22no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—42no—
79C23×C4—42no—
80C23×C4C22⋊C412yes—
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22×C411yescharacteristic
83C23×C4C2×D412yes—
84C2×M16—44no—
85C22⋊Q8—48no—
86C22⋊Q8—48no—
87#117—48no—
88#117—48no—
89C2×C4⋊C4—24no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—28no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—48no—
102C2×C4⋊C4C22⋊C412yes—
103C2×C4⋊C4C22⋊C412yes—
104C2×C4⋊C4C2×D412yes—
105C2×C4⋊C4C2×D412yes—
106C2×C22⋊C4—24no—
107C2×C22⋊C4—24no—
108C2×C22⋊C4—24no—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—44no—
112C2×C22⋊C4—42no—
113C2×C22⋊C4—42no—
114C2×C42—24no—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—42no—
118C2×C42—42no—
119C2×C42—42no—
120C2×C42—42no—
121C2×C42C22⋊C412yes—
122C2×C42C22⋊C412yes—
123C22⋊C8—44no—
124C22⋊C8—44no—
125#95—24no—
126#95—28no—
127#95—28no—
128#95—28no—
129C24—21no—
130C24—21no—
131C24—42no—
132C24—42no—
133C24—41no—
134C24—41no—
135C24#9912yes—
136C24C2×C22⋊C411yescharacteristic
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—84no—
144C2×Q8—84no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—24no—
150C22×C4—24no—
151C22×C4—24no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—48no—
164C22×C4—48no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—44no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—44no—
177C22×C4—42no—
178C22×C4—44no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4#9912yes—
186C22×C4C2×C22⋊C411yescharacteristic
187C22×C4C2×C22⋊C411yescharacteristic
188C22×C4#12014yes—
189M16—88no—
190C2×C8—84no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—48no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—416no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C22⋊C4—48no—
200C22⋊C4—48no—
201C22⋊C4—48no—
202C42—24no—
203C42—24no—
204C42—42no—
205C42—42no—
206C42—48no—
207C23—22no—
208C23—22no—
209C23—21no—
210C23—21no—
211C23—22no—
212C23—24no—
213C23—42no—
214C23—42no—
215C23—42no—
216C23—42no—
217C23—42no—
218C23—42no—
219C23—42no—
220C23—41no—
221C23—41no—
222C23—84no—
223C23—84no—
224C23#37911yescharacteristic
225C23C2×#9911yescharacteristic
226C23C2×#9911yescharacteristic
227Q8—84no—
228Q8—84no—
229C2×C4—24no—
230C2×C4—24no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—24no—
238C2×C4—48no—
239C2×C4—48no—
240C2×C4—48no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—44no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—44no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—44no—
251C2×C4—42no—
252C2×C4—44no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—82no—
260C2×C4—84no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4#41312yes—
268C2×C4#41312yes—
269C8—164no—
270C22—24no—
271C22—24no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—84no—
282C22—82no—
283C22—82no—
284C22—84no—
285C22#178111yescharacteristic
286C22#178511yescharacteristic
287C22#178511yescharacteristic
288C22#212612yes—
289C22#213312yes—
290C4—22no—
291C4—22no—
292C4—44no—
293C4—42no—
294C4—44no—
295C4—82no—
296C4—82no—
297C4—82no—
298C4—82no—
299C2—21no—
300C2—21no—
301C2—22no—
302C2—22no—
303C2—82no—
304C2#854811yescharacteristic
305C2#1269511yescharacteristic
306C2#1270211yescharacteristic
subgroupG/H
center, socleC22#1781
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊Q8 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #11512C2semidirect
3. #11511C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#24992
#115122
#115112
#85552